- #1

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[tex]S+\lambda =0[/tex] (Such that D=0)

[tex]S+2\lambda =0[/tex]

In case of a hyperbola, S is the pair of straight lines representing the asymptotes and [tex]\lambda[/tex] is any parameter.

My question is, are the first two equations the same? How would you find the equations of the asymptotes if you were given the equation of the curve.

The third equation is the conjugate hyperbola if [tex] S+\lambda =0[/tex] represents the original hyperbola. Is there any other way to find the conjugate hyperbola?

If [tex] \frac{x^2}{a^2} - \frac{y^2}{b^2} =1[/tex] is the equation of the original hyperbola, then does the equation [tex] \frac{x^2}{a^2} - \frac{y^2}{b^2} =-1[/tex] represent the conjugate hyperbola?